Given any complete, consistent extension of , we showed that there is a minimal model of . This model is unique up to (unique) isomorphism, it is rigid (i.e., it has no automorphisms other than the identity), and has no proper elementary substructures.

Given a model , let , the standard system of , be the set of those coded by elements of , where codes iff . Thus is the class of finite sets. We showed that if is nonstandard, contains all recursive sets, and that for any non-recursive there is a nonstandard such that .

We showed that Exponentiation is Diophantine, completing the proof of the Davis-Matiyasevich-Putnam-Robinson theorem. The result follows from a careful examination of certain second order linear recurrence relations.

Hilbert’s tenth problem asks whether there is an algorithm that given a polynomial with integer coefficients (in an arbitrary number of variables) determines whether it has integer roots. A celebrated theorem of Davis, Matiyasevich, Putnam and Robinson shows that this is not the case. Their result shows that the class of Diophantine sets coincides with the a priori larger class of r.e. (or ) sets.

We proved this result under the assumption that exponentiation is Diophantine. This is the key result, and will be dealt with in the following lecture.

Important update: In problem 4.(a), recall that is a universal predicate for unary formulas, so if is the Gödel number of a formula in one free variable , then holds iff holds. Hence, asking that is finite is the same as asking that is finite. Actually, this is a serious typo:

is -complete. The set is -complete.

Sorry about this. Either ignore 4.(a), or try to show (for extra credit) that the set is -complete, or (for a much more challenging problem) that the corresponding set with “cofinite” is -complete.

We defined the analog of the halting problem for any oracle and showed that any set r.e. in is many-to-one reducible to (in particular, it is recursive in ). Hence, is a complete set and no such set is recursive in .

We proved the –– (or index function) theorem and Kleene’s recursion (or fixed point) theorem. Finally, we introduced the notion of an index set and proved Rice’s theorem that the only recursive index sets are and .

We proved that the class of functions computable by means of Turing machines coincides with the class of recursive functions. Together with the characterization of the recursive functions as those admitting graphs, one can see this result as empirical evidence for Church’s thesis, the claim that the intitive notion of “algorithymically computable” is correctly formalized in the notion of recursive. An immediate consequence of this result is the existence of universal Turing machines.

We defined machines with oracles and stated the corresponding result: Given an oracle , a function is computable by a Turing machine with oracle iff is recursive in (meaning that it is built up from and the basic functions by composition, recursion and minimalization) iff has a graph definable in the structure . We then defined the partial order among subsets of which holds iff is recursive in .